Extended Kalman Filter for the Estimation and Fuzzy Optimal Control of Takagi-Sugeno Model

This chapter is aimed at improving the local and global approximation and modelling capability of Takagi-Sugeno (T-S) fuzzy model and the design of an optimal fuzzy controller. The main aim is obtaining high function approximation accuracy and fast convergence. The approach developed here can be considered as a generalized version of TS fuzzy identification method with optimized performance in estimating nonlinear functions. We propose an iterative method by applying the extended Kalman filter. We show that the Kalman filter is an effective tool in the estimation of T-S fuzzy model. It is a powerful mathematical tool for stochastic estimation from noisy environment. It has various applications in optimizing fuzzy systems. For example, it has been used to extract fuzzy rules from a given rule base (Wang, L. & Yen, J. (1998)) and to optimize the output function parameters of T-S fuzzy systems (Ramaswamy, P.; Edwards, R. R. & Lee, K. (1993)). For linear systems with white noise and measurement noise, the Kalman filter is known to be an optimal estimator. For nonlinear dynamic systems with coloured noise, the Kalman filter can be extended by linearizing the system around the current parameter estimates. This algorithm updates parameters in a way that is consistent with all previously measured data and generally converges in a few iterations. In this chapter, we describe how the extended Kalman filter can be applied to fuzzy system optimization. Fuzzy logic has been used to compute the gains of a bank of parallel Kalman filters in order to combine their outputs (Hsiao, C. (1999)). Fuzzy logic has also been used to tune the parameters of a Kalman filter ( Kobayashi, K.; Cheok, K. & Watanabe, K.(1995). ). A fuzzy controller (FC) based Linear Quadratic Regulator (LQR) is then proposed in order to show the effectiveness of the estimation method developed here in control applications. An Illustrative example of a highly nonlinear system is chosen to evaluate the robustness and remarkable performance of the proposed method. The main idea is to design a supervisory fuzzy controller capable to adjust the controller parameters in order to obtain the desired response. The reason behind this scheme is to combine the best features of fuzzy control and those of the optimal LQR. In control design, it is often of interest to design a controller to fulfil, in an optimal form, certain performance criteria and constraints in addition to stability. The theme of optimal control addresses this aspect of control system design. For linear systems, the problem of designing optimal controllers reduces to solving algebraic Riccati equations, which are usually easy to solve and detailed literature of their solutions can be found in many